Sounds like a hardware problem to me. The fact that the same problem happens with 10.1 and 10.2 versions of the software suggests hardware problems. BB10 devices don't connect to the BlackBerry infrastructure like older devices that used BIS. Now your data comes straight from your carrier, no special BlackBerry data plan needed.

Do you have a data plan with your carrier? You might call them and confirm that your account is provisioned properly. While you do not need a special BlackBerry data plan you still need some kind of data plan.

This could also be a network coverage issue. If a local cell tower is out of action you might not get a strong enough signal for data coverage. You might try updating your PRL (Preferred Roaming List).

The part that confuses me is that email quits working. If only apps that might use specific BlackBerry services quit working that would be one thing, but why does Browser and twitter app work but not email? Email is just email now.

If it were me I would do a clean OS reinstall and test. I know one would expect the OS upgrade to fix any OS glitches, but maybe not.